\section{3D Elements}
   \subsection{CTETRA Element (4 Nodes)}
    \[ m = \rho V \]
    \[ V = \frac{1}{6} |J| \]
    \[ [J] = \left[ \begin{array}{cccc}
               1   & 1   & 1   &   1 \\
               x_1 & x_2 & x_3 & x_4 \\
               y_1 & y_2 & y_3 & y_4 \\
               z_1 & z_2 & z_3 & z_4 \\
            \end{array}\right]\]
    \[ 6V = |J| = x_{21}(y_{23}z_{34} - y_{34}z_{23}) + x23(y_{34}z_{12}-y_{12}z_{34})+x_{43}(y_{12}z_{23})-y_{23}z_{12}\]
    \[ x_{ij} = x_i-x_j \]

    %The shape functions come from \cite{colorado16}
    \[ [N_i] = \frac{1}{6V} \left[ \begin{array}{cccc}
                6V_{01} & y_{42}z{32}-y{32}z_{42} & x_{32}z_{42}-x_{42}z_{32} & x_{42}y_{32}-x_{32}y_{42} \\
                6V_{02} & y_{42}z{32}-y{32}z_{42} & y_{43}z_{31}-x_{13}z_{34} & x_{31}y_{43}-x_{34}y_{13} \\
                6V_{03} & y_{42}z{32}-y{32}z_{42} & x_{14}z_{24}-x_{24}z_{14} & x_{24}y_{14}-x_{14}y_{24} \\
%                6V_{04} & y_{42}z{32}-y{32}z_{42} & x_{21}z_{13}-x_{31}z_{12} & x_{13}y_{21}-x_{12}y_{31} \]
            \end{array}\right]\]

    \[ 6V_{01} = x_2(y_3z_4-y_4z_3) + x_3(y_4z_2-y_2z_4)+x_4(y_2z_3-y_3z_2) \]
    \[ 6V_{02} = x_1(y_4z_3-y_3z_4) + x_3(y_1z_4-y_4z_1)+x_4(y_3z_1-y_1z_3) \]
    \[ 6V_{03} = x_1(y_2z_4-y_4z_2) + x_2(y_4z_1-y_1z_4)+x_4(y_1z_2-y_2z_1) \]
    \[ 6V_{04} = x_1(y_3z_2-y_2z_3) + x_2(y_1z_3-y_3z_1)+x_3(y_2z_1-y_1z_2) \]

    \[  [E] = \frac{E}{(1-2\nu)(1+\nu)} \left[ \begin{array}{cccccc}
               1   & \nu & \nu & 0   &   0             & 0 \\
               \nu & 1   & \nu & 0   &   0             & 0 \\
               \nu & \nu &   1 & 0   &   0             & 0 \\
               0   &   0 &   0 & \frac{1}{2}-\nu &   0 & 0 \\
               0   &   0 &   0 & 0   & \frac{1}{2}-\nu & 0 \\
               0   &   0 &   0 & 0   &   0             & \frac{1}{2}-\nu
            \end{array}\right] \]

    \[  [D] = \frac{E}{1-\nu^2} \left[ \begin{array}{ccc}
               1   & \nu & 0  \\
               \nu & 1   & 0 \\
               0   & 0   & \frac{1}{2} (1-\nu)  \\
            \end{array}\right] \]
    \[ [D] = [E] \]
    \[ [N] = \frac{1}{4} \left[ \begin{array}{c}
               (1+\eta_1)(1-\eta_2) \\
               (1+\eta_1)(1+\eta_2) \\
               (1-\eta_1)(1-\eta_2) \\
               (1-\eta_1)(1+\eta_2)
            \end{array}\right]\]
    \[ [dN_\eta1] = \frac{1}{4} \left[ \begin{array}{c}
               -\eta_2 + 1 \\
                \eta_2 + 1 \\
                \eta_2 - 1 \\
               -\eta_2 - 1
            \end{array}\right]\]
    \[ [dN_\eta2] = \frac{1}{4} \left[ \begin{array}{c}
              -1 - \eta_1 \\
               1 + \eta_1 \\
              -1 + \eta_1 \\
               1 - \eta_1
            \end{array}\right]\]
    
    \[ [e] = [B] [u_e] \]
    \[ [B] = \frac{1}{6V}\left[ \begin{array}{cccccccccccc}
               a1 & 0  & 0  & a2 & 0  & a3 & 0  & 0  & a4 & 0  & 0  \\
               0  & b1 & 0  & 0  & b2 & 0  & b3 & 0  & 0  & b4 & 0  \\
               0  & 0  & c1 & 0  & 0  & 0  & 0  & c3 & 0  & 0  & c4 \\
               b1 & a1 & 0  & b2 & a2 & b3 & a3 & 0  & b4 & a4 & 0  \\
               0  & c1 & b1 & 0  & c2 & 0  & c3 & b3 & 0  & c4 & b4 \\
               c1 & 0  & a1 & c2 & 0  & c3 & 0  & a3 & c4 & 0  & a4 
            \end{array}\right]\]
     \[  [K_e] = |J| [B]^T [D] [B] \]
     If the elastic moduli do not vary over the element:
     \[  [K_e] = V [B]^T [D] [B] \partial \eta_1 \partial \eta_12\partial \eta_3 \]
     \[ [u] = \left[ \begin{array}{cccccccccccc}
               u_{x1} & u_{y1} & u_{z1} & 
               u_{x2} & u_{y2} & u_{z2} & 
               u_{x3} & u_{y3} & u_{z3}
            \end{array}\right]^T\]
      \[ [K] = [T]^{-T} [K_e] [T]  \]
      
      \[ [M_e] = \int_V{\rho [N][N]^T dV} \]

   \subsection{CPENTA Element (6 Nodes)}
     \[ m = \rho V \]
     \[ A_1 = Area(p_1, p_2, p_3) \]
     \[ A_2 = Area(p_4, p_5, p_6) \]
     \[ p_{c1} = Centroid(p_1, p_2 ,p_3) \]
     \[ p_{c2} = Centroid(p_4, p_5 ,p_6) \]
     \[ V = (p_{c2}-p_{c1}) \frac{(A_2 - A_1)}{2} \]

   \subsection{CHEXA Element (8 Nodes)}
     \[ m = \rho V \]
     \[ A_1 = Area(p_1, p_2, p_3, p_4) \]
     \[ A_2 = Area(p_5, p_6, p_7, p_8) \]
     \[ p_{c1} = Centroid(p_1, p_2 ,p_3, p_4) \]
     \[ p_{c2} = Centroid(p_5, p_6 ,p_7, p_8) \]
     \[ V = (p_{c2}-p_{c1}) \frac{(A_2 - A_1)}{2} \]

    \[ [N] = \frac{1}{8} \left[ \begin{array}{c}
               (1-\eta_1)(1-\eta_2)(1-\eta_3) \\
               (1+\eta_1)(1-\eta_2)(1-\eta_3) \\
               (1+\eta_1)(1+\eta_2)(1-\eta_3) \\
               (1-\eta_1)(1+\eta_2)(1-\eta_3) \\
               (1-\eta_1)(1-\eta_2)(1+\eta_3) \\
               (1+\eta_1)(1-\eta_2)(1+\eta_3) \\
               (1+\eta_1)(1+\eta_2)(1+\eta_3) \\
               (1-\eta_1)(1+\eta_2)(1+\eta_3)
            \end{array}\right] \]
    \[ [N_{\eta1}] = \frac{1}{8} \left[ \begin{array}{c}
               -(1-\eta_2)(1-\eta_3) \\
                (1-\eta_2)(1-\eta_3) \\
                (1+\eta_2)(1-\eta_3) \\
               -(1+\eta_2)(1-\eta_3) \\
               -(1-\eta_2)(1+\eta_3) \\
                (1-\eta_2)(1+\eta_3) \\
                (1+\eta_2)(1+\eta_3) \\
               -(1+\eta_2)(1+\eta_3)
            \end{array}\right]\]
    \[ [N_{\eta2}] = \frac{1}{8} \left[ \begin{array}{c}
               -(1-\eta_1)(1-\eta_3) \\
               -(1+\eta_1)(1-\eta_3) \\
                (1+\eta_1)(1-\eta_3) \\
                (1-\eta_1)(1-\eta_3) \\
               -(1-\eta_1)(1+\eta_3) \\
               -(1+\eta_1)(1+\eta_3) \\
                (1+\eta_1)(1+\eta_3) \\
                (1-\eta_1)(1+\eta_3)
            \end{array}\right]\]
    \[ [N_{\eta3}] = \frac{1}{8} \left[ \begin{array}{c}
               -(1-\eta_1)(1-\eta_2) \\
               -(1+\eta_1)(1-\eta_2) \\
               -(1+\eta_1)(1+\eta_2) \\
               -(1-\eta_1)(1+\eta_2) \\
                (1-\eta_1)(1-\eta_2) \\
                (1+\eta_1)(1-\eta_2) \\
                (1+\eta_1)(1+\eta_2) \\
                (1-\eta_1)(1+\eta_2)
            \end{array}\right]\]

	This is a citation. \cite{bar}
        This is another.. \cite{colorado16}
        